Skip to main content
SpringerLink
Log in

On the Information Ratio of Non-perfect Secret Sharing Schemes

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beimel, A.: Secret-sharing schemes: a survey. In: Coding and Cryptology, Third International Workshop, IWCC 2011, Lecture Notes in Computer Science, vol. 6639, pp. 11–46 (2011)

  2. Beimel, A., Ben-Efraim, A., Padró, C., Tyomkin, I.: Multi-linear secret-sharing schemes. In: Theory of Cryptography, TCC 2014, Lecture Notes in Computer Science, vol. 8349, pp. 394–418 (2014)

  3. Beimel, A., Farràs, O., Mintz, Y.: Secret sharing schemes for very dense graphs. J. Cryptol. 29(2), 336–362 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beimel, A., Livne, N., Padró, C., Matroids can be far from ideal secret sharing. In: Theory of Cryptography, TCC 2008, Lecture Notes in Computer Science, vol. 4948, pp. 194–212 (2008)

  5. Beimel, A., Orlov, I.: Secret sharing and non-shannon information inequalities. IEEE Trans. Inform. Theory 57, 5634–5649 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benaloh, J., Leichter, J.: Generalized secret sharing and monotone functions. In: Advances in Cryptology, CRYPTO’88, Lecture Notes in Computer Science, vol. 403, pp. 27–35 (1990)

  7. Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: Proceedings of the ACM STOC’88, pp. 1–10 (1988)

  8. Blakley, G.R.: Safeguarding cryptographic keys. In: AFIPS Conference Proceedings, vol. 48, pp. 313–317 (1979)

  9. Blakley, G.R., Meadows, C.: Security of Ramp Schemes. In: Advances in Cryptology, CRYPTO’84, Lecture Notes in Computer Science, vol. 196, pp. 242–268 (1985)

  10. Bogdanov, A., Guo, S., Komargodski, I.: Threshold secret sharing requires a linear size alphabet. In: Electronic Colloquium on Computational Complexity, Report No. 131 (2016)

  11. Brickell, E.F.: Some ideal secret sharing schemes. J. Combin. Math. Combin. Comput. 9, 105–113 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Brickell, E.F., Davenport, D.M.: On the classification of ideal secret sharing schemes. J. Cryptol. 4, 123–134 (1991)

    MATH  Google Scholar 

  13. Capocelli, R.M., De Santis, A., Gargano, L., Vaccaro, U.: On the size of shares for secret sharing schemes. J. Cryptol. 6, 157–167 (1993)

    Article  MATH  Google Scholar 

  14. Cascudo, I., Cramer, R., Xing, C.: Bounds on the threshold gap in secret sharing and its applications. IEEE Trans. Inf. Theory 59, 5600–5612 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chaum, D., Crépeau, C., Damgård, I.: Multi-party unconditionally secure protocols. In: Proceedings of the ACM STOC’88, pp. 11–19 (1988)

  16. Chen, H., Cramer, R., de Haan, R., Cascudo Pueyo, I.: Strongly multiplicative ramp schemes, from high degree rational points on curves. In: Advances in Cryptology, EUROCRYPT 2008, Lecture Notes in Computer Science, vol. 4965, pp. 451–470 (2008)

  17. Chen, Q., Yeung, R.W.: Two-Partition-Symmetrical Entropy Function Regions. ITW 1–5 (2013)

  18. Cook, S.A., Pitassi, T., Robere, R., Rossman, B.: Exponential lower bounds for monotone span programs. In: Electronic Colloquium on Computational Complexity, Report No.64 (2016)

  19. Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley, New York (2006)

    MATH  Google Scholar 

  20. Cramer, R., Damgård, I., de Haan, R.: Atomic secure multi-party multiplication with low communication. In: Advances in Cryptology, EUROCRYPT 2008, Lecture Notes in Computer Science, vol. 4515, pp. 329–346 (2007)

  21. Cramer, R., Damgård, I., Maurer, U.: General secure multi-party computation from any linear secret-sharing scheme. In: Advances in Cryptology, EUROCRYPT 2000, Lecture Notes in Computer Science, vol. 1807, pp. 316–334 (2000)

  22. Csirmaz, L.: The size of a share must be large. J. Cryptol. 10, 223–231 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Csirmaz, L., Tardos, G.: Optimal information rate of secret sharing schemes on trees. IEEE Trans. Inform. Theory 59, 2527–2630 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Desmedt, Y.: Threshold cryptography. Euro. Trans. Telecommun. 5, 449–457 (1994)

    Article  Google Scholar 

  25. Farràs, O.: Recent advances in non-perfect secret sharing schemes. In: Conference on Computability in Europe, CiE 2016. Lecture Notes in Computer Science, vol. 9709, pp. 89–98 (2016)

  26. Farràs, O., Hansen, T., Kaced, T., Padró, C.: Optimal non-perfect uniform secret sharing schemes. In: Advances in Cryptology, CRYPTO 2014, Lecture Notes in Computer Science, vol. 8617, pp. 217–234 (2014)

  27. Farràs, O., Martín, S., Padró, C.: A note on ideal non-perfect secret sharing schemes. Cryptology ePrint Archive 2016/348 (2016)

  28. Farràs, O., Metcalf-Burton, J.R., Padró, C., Vázquez, L.: On the optimization of bipartite secret sharing schemes. Des. Codes Cryptogr. 63, 255–271 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Farràs, O., Padró, C.: Extending Brickell–Davenport theorem to non-perfect secret sharing schemes. Des. Codes Cryptogr. 74(2), 495–510 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Franklin, M., Yung, M.: Communication Complexity of Secure Computation, STOC 1992, pp. 699–710 (1992)

  31. Fujishige, S.: Polymatroidal dependence structure of a set of random variables. Inf. Control 39, 55–72 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fujishige, S.: Entropy functions and polymatroids–combinatorial structures in information theory. Electron. Comm. Jpn. 61, 14–18 (1978)

    MathSciNet  Google Scholar 

  33. Ishai, Y., Kushilevitz, E., Strulovich, O.: Lossy Chains and Fractional Secret Sharing. In: STACS 2013, LIPICS, vol. 20, pp. 160–171 (2013)

  34. Ito, M., Saito, A., Nishizeki, T.: Secret sharing scheme realizing any access structure. In: Proceedings of the IEEE Globecom’87, pp. 99–102 (1987)

  35. Jackson, W.-A., Martin, K.M.: Geometric secret sharing schemes and their duals. Des. Codes Cryptogr. 4, 83–95 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kaced, T.: Almost-perfect secret sharing. In: Proceedings of 2011 IEEE International Symposium on Information Theory, ISIT 2011, pp. 1603–1607 (2011). Full version available at arXiv:1103.2544

  37. Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Trans. Inform. Theory 29, 35–41 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kothari, S.C.: Generalized linear threshold scheme. In: Advances in Cryptology, CRYPTO’84, Lecture Notes in Computer Science, vol. 196, pp. 231–241 (1985)

  39. Kurosawa, K., Okada, K., Sakano, K., Ogata, W., Tsujii, S.: Nonperfect secret sharing schemes matroids. In: Advances in Cryptology, EUROCRYPT 1993, Lecture Notes in Computer Science, vol. 765, pp. 126–141 (1994)

  40. Martí-Farré, J., Padró, C.: On secret sharing schemes matroids and polymatroids. J. Math. Cryptol. 4, 95–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Martín, S., Padró, C., Yang, A.: Secret sharing inequalities, rank, inequalities, information. In: Advances in Cryptology, CRYPTO 2013, Lecture Notes in Computer Science, vol. 8043, pp. 277–288 (2012)

  42. Massey, J.L.: Minimal codewords and secret sharing. In: Proceedings of the Sixth Joint Swedish–Russian Workshop on Information Theory, Molle, Sweden, August 1993, pp. 269–279 (1993)

  43. McEliece, R.J., Sarwate, D.V.: On sharing secrets and Reed–Solomon codes. Commun. ACM 24, 583–584 (1981)

    Article  MathSciNet  Google Scholar 

  44. Ogata, W., Kurosawa, K., Tsujii, S.: Nonperfect secret sharing schemes. In: Advances in Cryptology, Auscrypt 92, Lecture Notes in Computer Science, vol. 718, pp. 56–66 (1993)

  45. Okada, K., Kurosawa, K.: Lower bound on the size of shares of nonperfect secret sharing schemes. In: Advances in Cryptology, Asiacrypt 94, Lecture Notes in Computer Science, vol. 917, pp. 33–41 (1995)

  46. Oxley, J.G.: Matroid Theory. The Clarendon Press, New York (1992)

    MATH  Google Scholar 

  47. Padró, C.: Lecture Notes in Secret Sharing. Cryptology ePrint Archive 2012/674

  48. Padró, C., Vázquez, L., Yang, A.: Finding lower bounds on the complexity of secret sharing schemes by linear programming. Discrete Appl. Math. 161, 1072–1084 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. Paillier, P.: On ideal non-perfect secret sharing schemes. In: Security Protocols, 5th International Workshop, Lecture Notes in Computer Science, vol. 1361, pp. 207–216 (1998)

  50. Schrijver, A.: Combinatorial Optimization Polyhedra and Efficiency. Springer, Berlin (2003)

    MATH  Google Scholar 

  51. Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  52. Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)

    MATH  Google Scholar 

  53. Yoshida, M., Fujiwara, T.: Secure construction for nonlinear function threshold ramp secret sharing. In: IEEE International Symposium on Information Theory, ISIT 2007, pp. 1041–1045 (2007)

  54. Yoshida, M., Fujiwara, T., Fossorier, M.: Optimum general threshold secret sharing. In: Security, Information Theoretic, ICITS 2012, Lecture Notes in Computer Science, vol. 7412, pp. 187–204 (2012)

Download references

Author information

Authors and Affiliations

  1. Universitat Rovira i Virgili, Tarragona, Spain

    Oriol Farràs

  2. Royal Holloway, University of London, London, UK

    Torben Brandt Hansen

  3. LACL, UPEC, Université de Paris-Est, Paris, France

    Tarik Kaced

  4. Universitat Politècnica de Catalunya, Barcelona, Spain

    Carles Padró

Authors
  1. Oriol Farràs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Torben Brandt Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tarik Kaced
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Carles Padró
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oriol Farràs.

Additional information

Part of this work was presented in the conference CRYPTO 2014 and appeared in its proceedings [26]. Oriol Farràs is supported by the Spanish Government through a Juan de la Cierva grant, TIN2011C27076-C03-01, TIN2014-57364-C2-1-R, by the European Union through H2020-ICT-2014-1-644024, and by the Government of Catalonia through Grant 2014 SGR 537. Part of the work of Torben B. Hansen was done while at Aarhus University and Universitat Rovira i Virgili. Tarik Kaced is supported in part by a grant from the University Grants Committee of the Hong Kong SAR, China (Project No. AoE/E-02/08), and by EQINOCS ANR 11 BS02 004 03. Carles Padró is supported by the Spanish Government under the project MTM2013-41426-R. Part of this research work was done while Carles Padró was with Nanyang Technological University, Singapore.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Farràs, O., Hansen, T.B., Kaced, T. et al. On the Information Ratio of Non-perfect Secret Sharing Schemes. Algorithmica 79, 987–1013 (2017). https://doi.org/10.1007/s00453-016-0217-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-016-0217-9

Keywords

Search

Navigation